Dynamical instabilities and anomalous transport in gauge theories

Dynamical instabilities and anomalous transport in gauge theories with chiral fermions Pavel Buividovich (Regensburg)

Some recent real-time studies • Real-time dynamics of chirality pumping [1611. 05294] with Semen Valgushev • Chiral plasma instability and inverse cascade [1509. 02076] with Maksim Ulybyshev • Evaporating black holes from holography and Super-Yang. Mills, with M. Hanada [in preparation]

Why chiral plasma? Collective motion of chiral fermions • High-energy physics: ü Quark-gluon plasma ü Hadronic matter ü Neutrinos/leptons in Early Universe • Condensed matter physics: ü Liquid Helium (!!!) [G. Volovik] ü Weyl semimetals ü Topological insulators

Anomalous transport: Hydrodynamics Classical conservation laws for chiral fermions • Energy and momentum • Angular momentum • Electric charge No. of left-handed • Axial charge No. of right-handed Hydrodynamics: • Conservation laws • Constitutive relations Axial charge violates parity New parity-violating transport coefficients

Anomalous transport: CME, CSE, CVE Chiral Magnetic Effect [Kharzeev, Warringa, Fukushima] Chiral Separation Effect [Zhitnitsky, Son] Chiral Vortical Effect [Erdmenger et al. , Teryaev, Banerjee et al. ] Flow vorticity Origin in quantum anomaly!!!

Chiral anomaly: brief intro Charged particle in magnetic field: 1 D motion (transverse motion “confined”, spin || magnetic field) Massless fermions have two “chiral states” Spin (anti)parallel to momentum Quantum field theory: Vacuum is full …of virtual particles Let’s switch electric field!!!

Chiral anomaly: brief intro e+ E e- Right-moving particles Left-moving anti-particles

Chiral anomaly: going 3 D In quantum theory 1 D reduction = lowest Landau level Number of LLLs/Area = B/(2π) Anomaly equation π0 γ γ π0

Chiral Magnetic Effect Imbalance of left- and right-movers μA -μ A Electric current In 3 D: direction of B + Landau level degeneracy … In fact, “rotation” of anomaly diagram

How to magnetize and rotate chiral matter? Relative motion of two large charges (Z ~ 100) Large magnetic field in the collision region URQMD simulations Au+Au No backreaction From [Skokov, Toneev, Ar. Xiv: 0907. 1396] Weak energy dependence!!!

Before hydro / kinetics … Glasma state E || B Production of axial charge! Magnetic field is maximal! Key objectives: 1) Initial conditions for anomalous hydro (axial charge density, electric current, energy density) 2) Real-time shocks, plasma instabilities, etc. Key tools: Classical-statistical field theory: fully quantum CHIRAL fermions+ classical gauge fields Key innovation [PB+coauth, 1509. 02076, 1611. 05294]

Interplay of gauge and fermion chirality μA, QA- not “canonical” charge/chemical potential “Conserved” charge: Chern-Simons term (Magnetic helicity) Integral gauge invariant (without boundaries)

Chirality pumping [With S. Valgushev] What happens with this equation if gauge [Anselm, fields are dynamical? Iohansen’ 88] Q 5(t) External Dynamic t

Chirality pumping in E || B Maxwell equations with Bz=const, Ez(z, t), other comps. 0 Anomaly equation Vector and axial currents (Lowest Landau level = 1 D fermions)

Chirality pumping in E || B Combining everything Massive mode in chiral plasma (Chiral Magnetic Wave mixing with plasmon) Mass is the back-reaction effect (Also corrections due to higher Landau levels)

Chirality pumping in E || B Now assume Ez(z, t)=Ez(t) No backreaction vs. Backreaction Q 5(t) No backreaction (ext. fields) Backreaction (dynamic fields) t Max. axial charge ~ B 1/2 vs B Times > B-1/2

Chiral instability and Inverse cascade Energy of large-wavelength modes grows … at the expense of short-wavelength modes! • Generation of cosmological magnetic fields [Boyarsky, Froehlich, Ruchayskiy, 1109. 3350] • Circularly polarized, anisotropic soft photons in heavy-ion collisions [Hirono, Kharzeev, Yin 1509. 07790][Torres-Rincon, Manuel, 1501. 07608 • Spontaneous magnetization of topological insulators [Ooguri, Oshikawa, 1112. 1414] • THz circular EM waves from Dirac/Weyl semimetals [Hirono, Kharzeev, Yin 1509. 07790]

Anomalous Maxwell equations (now a coarse approximation…) Maxwell equations + ohmic conductivity + CME Ohmic conductivity Chiral magnetic conductivity Conservation of energy in the presence of CME Keeping chiral imbalance constant requires work

Anomalous Maxwell equations (now a coarse approximation…) • Assume dispersionless electric conductivity and CME response • Neglect nonlinear effects (Schwinger pair production, etc) Plane wave solution

Chiral plasma instability Dispersion relation At k < = μA/(2 π2): Im(w) < 0 Unstable solutions!!! Cf. [Hirono, Kharzeev, Yin 1509. 07790] Real-valued solution:

Chiral instability and Inverse cascade Energy of large-wavelength modes grows … at the expense of short-wavelength modes! 2 D turbulence, from H. J. H. Clercx and G. J. van Heijst Appl. Mech. Rev 62(2), 020802

Helical structure of unstable solutions Helicity only in space – no running waves E || B - ``topological’’ density Note: E || B not possible for oscillating ``running wave’’ solutions, where E • B=0 What can stop the instability?

What can stop the instability? For our unstable solution with μA>0: Instability depletes QA μA and chi decrease, instability stops Energy conservation: Keeping constant μA requires work!!!

Real-time dynamics of chiral plasma • • - Approaches used so far: Anomalous Maxwell equations Hydrodynamics (long-wavelength) Holography (unknown real-world system) Chiral kinetic theory (linear response, relaxation time, long-wavelength…) What else can be important: Nontrivial dispersion of conductivities Developing (axial) charge inhomogeneities Nonlinear responses Let’s try to do numerics!!!

Real-time simulations: classical statistical field theory approach [Son’ 93, Aarts&Smit’ 99, J. Berges&Co] • • • Full real-time quantum dynamics of fermions Classical dynamics of electromagnetic fields Backreaction from fermions onto EM fields Vol X Vol matrices, Bottleneck for numerics!

Decay of axial charge and inverse cascade Excited initial [PB, Ulybyshev 1509. 02076] state with μA < 1 on the lattice chiral imbalance 2 Hamiltonian is CP-symmetric, State is not!!! [No momentum separation] To reach k < μA/(2 π ): • 200 x 20 lattices, Wilson fermions, MPI parallelism • Translational invariance in 2 out of 3 dimensions To detect instability and inverse cascade: • Initially n modes of EM fields with equal energies and random polarizations

Axial charge decay 200 x 20 lattice, μA= 0. 75 If amplitude small, no decay

Universal late-time scaling [Yamamoto 1603. 08864], [Hirono, Kharzeev, Yin 1509. 07790] QA ~ -1/2 t

Power spectrum and inverse cascade Fourier transform the fields Basis of helical components Smearing the short-scale fluctuations

Power spectrum and inverse cascade (L) Magnetic (R) Small amplitude, QA ~ const Electric Chiral laser with circular polarization!

Power spectrum and inverse cascade (L) Magnetic (R) Electric Large amplitude, large μA, QA decays

Power spectrum and inverse cascade (L) Magnetic (R) Electric Large amplitude, QA decays

Discussion and outlook • Axial charge decays with time (nature doesn’t like fermion chirality) Large-scale helical EM fields Short EM waves decay Non-linear mechanism! Instability stops much earlier than predicted by anomalous Maxwell eqns. !!! Dispersionless linear response not so good • •

Chiral instability: discussion and outlook Outlook • Premature saturation of instability: physics or artifact of Wilson fermions? • Overlap fermions in CSFT?

Real-time instabilities of gauge fields + fermions: interesting side development Black hole dynamics N=1 Supersymmetric Yang-Mills in D=1+9: gauge bosons+adjoint Majorana-Weyl fermions Reduce to a single point = BFSS matrix model [Banks, Fischler, Shenker, Susskind’ 1997] N x N hermitian matrices Majorana-Weyl fermions, N x N hermitian matrices Recent numerics: [Bergner, Liu, Wenger, 1612. 04291]

BFSS model and “holography” “Holographic” dictionary [Witten’ 96]: • Xiiμ = D 0 brane positions • Xijμ = open string excitations System of N D 0 branes joined by open strings Eigenvalues of X 2 = “radial coordinates”

Motivation Classical equations of motion are chaotic [Saviddy; Berenstein; Susskind; Hanada; …] • Positive Lyapunov exponents • Xμ matrices forget initial conditions • Kolmogorov entropy produced Stringy interpretation: black hole formation

• • • Expectation So far classical real-time simulations Classical flat directions [Xμ, Xν]=0 entropically suppressed Fermions enhance tunneling (repulsive force) Quantum fluctuations of bosons cancel repulsion at large distances (SUSY) D 0 branes tunneling = (? ? ? ) Black hole evaporation

Classical-statistical field theory (CSFT) [Son, Aarts, Smit, Berges, Tanji, Gelis, …] - Schwinger pair production - Axial charge generation in glasma - (Chiral) plasma instabilities Closed system of equations: Numerical solution! Fermions are costly! CPU time + RAM memory scaling ~ N 4 Parallelization is necessary

Initial conditions: some excited state … for nontrivial evolution (thinking about black holes, we still believe in quantum mechanics) Our initial conditions: Gaussian random, dispersion f Gaussian random, dispersion 1 Fermions are in ground state at given Xai Tricky for Majoranas, no Dirac sea!!!

Some results N = 8, f = 0. 48 Hawking radiation of D 0 branes 2 X Thermalization With fermions Classical t

Some results N = 8, f = 0. 48 Not all configurations evaporate 2 X Classical With fermions t

Constant acceleration at late times 2 X t • • • Boson kinetic energy grows without bound Fermion energy falls down Origin of const force – SUSY violation?

CSFT and SUSY In full quantum theory Fierz identity (cyclic shift of indices): In CSFT approximation Fermionic 3 pt function seems necessary!

Going beyond CSFT • Return to Heisenberg eqs of motion • VEV with more non-trivial correlators First, test this idea on the simplest example Tunnelling between potential wells? Absent in CSFT!

Next step: Gaussian Wigner function Assume Gaussian wave function at any t Simpler: Gaussian Wigner function For other correlators: use Wick theorem! Derive closed equations for x, p, σxx , σxp , σpp

Origin of tunnelling Positive force even at x=0 (classical minimum) Quantum force causes classical trajectory to leave classical minimum

Some results N = 8, f = 0. 48 σ2 = 0. 0, 0. 1, 0. 2, 1. 0 + no fermions

Some results N = 8, f = 0. 48 σ2 = 0. 0, 0. 1, 0. 2, 1. 0 + no fermions

Evaporation threshold: N = 6, f = 0. 50 σ2 = 0. 0, 0. 2, 1. 0 + no fermions

Quantum fluctuations vs evaporation Quantum fluctuations of X variables suppress evaporation!!! • Small σ2: evaporation becomes slower • Intermediate σ2: no evaporation • Large σ2: fermions negligible • Expected trend, but still no exact cancellation of bosons and fermions • Fine-tuning with realistic initial conditions? What effect quantum fluctuations of X have on classical dynamics?

Brief summary • Chirality pumping: backreaction makes axial charge and CME current oscillating, QA~B 1/2 scaling vs. QA ~ B • Chiral plasma instability stops earlier than chiral imbalance is depleted • Real-time simulations of black hole formation and evaporation Thank you for your attention!!!